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# A generalized framework of linear multivariable control

Author

Publisher

Butterworth-Heinemann is an imprint of Elsevier

Publication Date

2017.

Language

English

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ISBN

9780081019474

## Table of Contents

#### From the eBook

Front Cover; A Generalized Framework of Linear Multivariable Control; Copyright; Contents; Chapter 1: Introduction; Chapter 2: Mathematical preliminaries; 2.1 Vector algebra; 2.2 Matrix algebra; 2.2.1 Matrix properties; 2.2.2 Basic matrix operations; 2.3 Matrix inverse; 2.4 Solving system of linear equation; 2.4.1 Gauss method; 2.4.2 A general scheme for solving system of linear equation; 2.5 Linear differential equation; 2.5.1 Introduction; 2.5.2 Homogeneous equations with constant coefficients; 2.5.3 Nonhomogeneous equation with constant coefficients.

2.5.4 Equation with variable coefficients2.5.5 Systems of linear differential equations; 2.6 Matrix differential equation; 2.6.1 Introduction; 2.6.2 Stability and steady state of the matrix system; 2.6.3 Solution in matrix form; 2.6.4 Solving matrix ordinary differential equations; 2.7 Laplace transform; 2.7.1 Introduction; 2.7.2 Formal definition; 2.7.3 Region of convergence; 2.7.4 Laplace transform pair table; 2.7.5 Properties and theorems; 2.7.6 Inverse Laplace transform; Chapter 3: Generalized inverse of matrix and solution of linear system equation; 3.1 The generalized inverse of matrix.

3.1.1 The left inverse and right inverse3.1.2 Moore-Penrose inverse; 3.1.3 The minimization approach to solve an algebraic matrix equation; 3.2 The full rank decomposition theorem; 3.3 The least square solution to an algebraic matrix equation; 3.3.1 The solution to the compatible linear equations; 3.3.2 The least square solution of incompatible equation; 3.3.3 The minimum norm least squares solution for the equations; 3.4 The singular value decomposition; Chapter 4: Polynomial fraction description; 4.1 Introduction; 4.2 Right polynomial fractions; 4.3 Left polynomial fraction.

4.4 Column and row degrees4.5 Minimal realization; 4.6 Poles and zeros; 4.7 State feedback; Chapter 5: Stability; 5.1 Internal stability; 5.1.1 Uniform exponential stability; 5.1.2 Uniform asymptotic stability; 5.1.3 Lyapunov transformation; 5.2 Lyapunov stability; 5.2.1 Introduction; 5.2.2 Uniform stability; 5.2.3 Uniform exponential stability; 5.2.4 Instability; 5.2.5 Time-invariant case; 5.3 Input-output stability; 5.3.1 Uniform bounded-input bounded-output stability; 5.3.2 Relation to uniform exponential stability; 5.3.3 Time-invariant case.

Chapter 6: Fundamental approaches to control system analysis6.1 PMD theory of linear multivariable control systems; 6.2 Behavioral approach in systems theory; 6.3 Chain-scattering representations; 6.4 Conclusions; Chapter 7: Determination of finite and infinite frequency structure of a rational matrix; 7.1 Introduction; 7.2 The Toeplitz rank information; 7.3 To determine the Smith form of a polynomial matrix; 7.4 To determine the Smith-McMillan form at infinity of a rational matrix; 7.5 To determine the Smith-McMillan form of a rational matrix; 7.6 Conclusions.

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